Adaptive Optimal Scaling of Metropolis-Hastings Algorithms Using the Robbins-Monro Process

We present an adaptive method for the automatic scaling of Random-Walk Metropolis-Hastings algorithms, which quickly and robustly identifies the scaling factor that yields a specified overall sampler acceptance probability. Our method relies on the use of the Robbins-Monro search process, whose performance is determined by an unknown steplength constant. We give a very simple estimator of this constant for proposal distributions that are univariate or multivariate normal, together with a sampling algorithm for automating the method. The effectiveness of the algorithm is demonstrated with both simulated and real data examples. This approach could be implemented as a useful component in more complex adaptive Markov chain Monte Carlo algorithms, or as part of automated software packages

A Comparative Review of Dimension Reduction Methods in Approximate Bayesian Computation

Approximate Bayesian computation (ABC) methods make use of comparisons between simulated and observed summary statistics to overcome the problem of computationally intractable likelihood functions. As the practical implementation of ABC requires computations based on vectors of summary statistics, rather than full data sets, a central question is how to derive low-dimensional summary statistics from the observed data with minimal loss of information. In this article we provide a comprehensive review and comparison of the performance of the principal methods of dimension reduction proposed in the ABC literature. The methods are split into three nonmutually exclusive classes consisting of best subset selection methods, projection techniques and regularization. In addition, we introduce two new methods of dimension reduction. The first is a best subset selection method based on Akaike and Bayesian information criteria, and the second uses ridge regression as a regularization procedure. We illustrate the performance of these dimension reduction techniques through the analysis of three challenging models and data sets.Comment: Published in at http://dx.doi.org/10.1214/12-STS406 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org